Search

We present a generator matrix of the form $ [ \sigma(v_1) \ | \ \sigma(v_2)] $, where $ v_1 \in RG $ and $ v_2\in RH $, for finite groups $ G $ and $ H $ of order $ n $ for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring $ R $. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve $ 10 $ of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain $ 82 $ new binary linear complementary dual codes, $ 50 $ of which are either optimal or near optimal of lengths $ 41 \leq n \leq 61 $ which are new to the literature. [ABSTRACT FROM AUTHOR]

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group Z p e . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with p ≡ 1 (mod 4) and e is odd with e > 1 ; and all lengths that are 0 (mod 4) when p is an odd prime with p ≡ 3 (mod 4) and e is odd with e > 1. [ABSTRACT FROM AUTHOR]

In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $ k^{th} $-range neighbours, and search for binary $ [72, 36, 12] $ self-dual codes. In particular, we present six generator matrices of the form $ [I_{36} \ | \ \tau_6(v)], $ where $ I_{36} $ is the $ 36 \times 36 $ identity matrix, $ v $ is an element in the group matrix ring $ M_6(\mathbb{F}_2)G $ and $ G $ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $ [72, 36, 12] $ self-dual codes directly over the finite field $ \mathbb{F}_2 $. We construct 1471 new Type I binary $ [72, 36, 12] $ self-dual codes with the rare parameters $ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $ in their weight enumerators. [ABSTRACT FROM AUTHOR]

Construction of maximal entanglement-assisted quantum error correction (EAQEC) codes is one of the fundamental problems of quantum computing and quantum information. The objective of this paper is twofold: firstly, to obtain all possible construction matrices of the linear codes over the skew group ring F 4 ⋊ φ G , where G is the cyclic and dihedral groups of finite orders; and secondly, to obtain some good maximal EAQEC codes over the finite field F 4 by using skew construction matrices. Additionally, to speed up the computational search time, we employ a nature inspired heuristic optimisation algorithm, the virus optimisation (VO) algorithm. With our method, we obtain a number of good maximal EAQEC codes over the finite field F 4 in a reasonably short time. In particular, we improve the lower bounds of 18 maximal EAQEC codes that exist in the literature. Moreover, some of our EAQEC codes turn out to be also maximum distance separable (MDS) codes. Also, by using our construction matrices, we provide counterexamples to Theorems 4 and 5 of Lai et al. (Quantum Inf Process 13(4):957–990, 2014), on the non-existence of maximal EAQEC codes with parameters [ [ n , 1 , n ; n - 1 ] ] and [ [ n , n - 1 , 2 ; 1 ] ] for an even length n. We also give a counterexample to another Theorem found in Lai and Ashikhmin (IEEE Trans Inf Theory 64:(1), 622–639, 2018), which states that there is no entanglement-assisted stabilizer code with parameters [ [ 4 , 2 , 3 ; 2 ] ] 4. [ABSTRACT FROM AUTHOR]

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters γ = 7 , 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction [ABSTRACT FROM AUTHOR]

In this work, we present a number of generator matrices of the form [ I 2 n | τ 2 (v) ] , where I 2 n is the 2 n × 2 n identity matrix, v is an element in the group matrix ring M 2 (R) G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [ 72 , 36 , 12 ] self-dual codes directly over the finite field F 2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings. [ABSTRACT FROM AUTHOR]